Homogeneous Designs and Geometric Lattices

1. IN~~DUCTION During the last 20 years, there has been a great deal of research concerning designs with A= 1 admitting 2-transitive groups. The following theorems will be proved in this note; they are fairly simple consequences of the classification ' of all finite simple groups (see, e.g., [6]). THEOREM 1. Let 9 be a design with ,! = 1 admitting an automorphism group 2-transitive on points. Then 9 is one of the following designs: (i) PG(d qh (ii) AG(d, q), (iii) The design with u=q3 + 1 and k=q+ 1 associated with PSU(3, q) or *G,(q), (iv) One of two affine planes, having 34 or 36 points [S, p. 2361, or (v) One of two designs having u = 36 and k = 3' [12]. THEOREM 2. Let Y be a finite geometric lattice of rank at least 3 such that Aut 6p is transitive on ordered bases. Then either (i) Y is a truncation of a Boolean lattice or a projective or affine geometry, (ii) 9 is the lattice associated with a (iii) 6p is the lattice associated with the 65-point design for PSU(3, 4). The groups in Theorems 1 and 2 are described in the course of the proof. It would, of course, be desirable to have more elementary proofs of both ' At the time of writing (December 1982), this classification is not quite complete: the uniqueness of the Monster has not been proved. However, this does not cause any dilliculties with our use of the classification.